The Battenberg/Mountbatten link is more direct than a distant relation to a possibly illegitimate son of the Prince of Hesse.
You can find this on the web: Queen Elizabeth II confirmed the royal Windsor name in a declaration following her accession in 1952. But in 1960 Queen Elizabeth II and her husband Prince Philip announced yet another name change. Prince Philip of Greece and Denmark, whose mother had been Alice of Battenberg, had already Anglicized his name to Philip Mountbatten when he married Elizabeth in 1947. (Interestingly, all four of Philip’s sisters, all now deceased, married Germans.) In her 1960 declaration to the Privy Council, the Queen expressed her wish that her children by Philip (other than those in line for the throne) would henceforth bear the hyphenated name Mountbatten-Windsor. The royal family’s name remained Windsor.
The Mountbatten “name change” from the Germanic thus relates directly to royal family.
The point being that to pledge loyalty to the Queen of England is not a loyalty to Britain but to its royal family. That family is less of British blood than many Britons.
But at the time the name was changed from Battenberg to Mountbatten, Prince Philip wasn't even a gleam in his daddy's eye. His mother was a daughter of Prince Louis and a great-granddaughter of Queen Victoria.
He was the most English of kings, and so was his son George VI.